Resume:

A theorem of Karpelevich-Mostow from 1950s states that every continuous isometric PO(1,n)-actions on an m-dimensional hyperbolic space preserves an n-dimensional subspace, thus there is no continuous irreducible representations of PO(1,n) in PO(1,m). But if m=∞, using a PO(1,n)-action on the unit n-sphere, the Delzant-Monod-Py construction provides a familly of non-totally geodesic quasi-isometric embedding of n-dimensional hyperbolic space into the infinite dimensional one, and these embeddings will induce continuous irreducible representations of PO(1,n) in PO(1,∞). These representations are called exotic. In this talk, we will use the Debin-Fillastre-Long hyperbolic model for convex bodies to give a different construction of non-totally geodesic quasi-isometric embedding of the hyperbolic plane in the infinite dimensional hyperbolic space, which will also induce an exotic representation of PSL(2,R). With the classical results from convex geometry, we show that this exotic representation is convex-cocompact, and that the quotient of the minimal PSL(2,R)-invariant closed convex set by its action is homeomorphic to the oriented Banach-Mazur compactum. Moreover, we also compute the dimension of this convex set and the Hausdorff dimension of the PSL(2,R)-limit set. This is a joint work with François Fillastre and David Xu.